Problem: Integrate. $\int\left(-\dfrac1x+7e^x \right)dx=\,?$ Choose 1 answer: Choose 1 answer: (Choice A) A $-\ln(x)+7e^x+C$ (Choice B) B $-\ln(x)+e^{7x}+C$ (Choice C) C $-\ln|x|+7e^x+C$ (Choice D) D $-\ln|x|+e^{7x}+C$
Answer: We can integrate using the following formulas for the indefinite integrals of $e^x$ and $\dfrac1x$ : $\begin{aligned} &\int e^x\,dx=e^x+C \\\\ &\int \dfrac1x\,dx=\ln|x|+C \end{aligned}$ $\begin{aligned} &\phantom{=}\int\left(-\dfrac1x+7e^x \right)dx \\\\ &=-1\int \dfrac1x\,dx+7\int e^x \,dx \\\\ &=-\ln|x|+7e^x+C \end{aligned}$